However, when the constraint \({\cal F}\)  in (REF ) is \({\cal F}_2\)  or \({\cal F}_3\) , the simplified subproblem \({\hat{\cal P}_{ \rm passive}}\)  in (REF ) becomes non-convex. The common solution to address such problem is to relax the constraint of the phase shift matrix in \({\cal F}_2\)  or \({\cal F}_3\)  [1]}, [2]}. Hence, by introducing an auxiliary variable \(\bf {\tilde{\Theta }}\) , the simplified subproblem \({\hat{\cal P}_{ \rm passive}}\)  in (REF ) can be relaxed as
\(\begin{aligned}{\hat{\cal P}_{ \text{ passive-relaxed}}:}~&\mathop {{\rm {max}}}\limits _{\bf {\tilde{\Theta }}}~{{g}_7}({\mathbf { \tilde{\Theta }}}) = - {{\mathbf {\tilde{\theta }}}^H}{\bf {\Lambda }}{\mathbf { \tilde{\theta }}} +{\mathfrak {R}} \left\lbrace 2 {{\mathbf {\tilde{\theta }}}^H}{\mathbf {\nu }}\right\rbrace - \zeta \\&~{\rm {s.t.}}~~~C_2:{{\tilde{\theta }_{r,j}}}\in {\cal F}_1,\forall r\in {{\cal R}},\forall j\in {{\cal N}}.\end{aligned}\) 
